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                <h2 id="「力扣」第-34-题：排序数组查找首、末位置（简单）"><a href="#「力扣」第-34-题：排序数组查找首、末位置（简单）" class="headerlink" title="「力扣」第 34 题：排序数组查找首、末位置（简单）"></a>「力扣」第 34 题：排序数组查找首、末位置（简单）</h2><ul>
<li><a href="https://leetcode-cn.com/problems/find-first-and-last-position-of-element-in-sorted-array/" target="_blank" rel="noopener">链接</a></li>
<li><a href="https://leetcode-cn.com/problems/find-first-and-last-position-of-element-in-sorted-array/solution/si-lu-hen-jian-dan-xi-jie-fei-mo-gui-de-er-fen-cha/" target="_blank" rel="noopener">题解链接（含视频题解）</a></li>
</ul>
<blockquote>
<p>给定一个按照升序排列的整数数组 <code>nums</code>，和一个目标值 <code>target</code>。找出给定目标值在数组中的开始位置和结束位置。</p>
<p>你的算法时间复杂度必须是 $O(\log n)$ 级别。</p>
<p>如果数组中不存在目标值，返回 <code>[-1, -1]</code>。</p>
<p>示例 1：</p>
<pre><code>输入: nums = [5, 7, 7, 8, 8, 10], target = 8
输出: [3, 4]</code></pre><p>示例 2：</p>
<pre><code>输入: nums = [5, 7, 7, 8, 8, 10], target = 6
输出: [-1, -1]</code></pre></blockquote>
<h3 id="思路"><a href="#思路" class="headerlink" title="思路"></a>思路</h3><blockquote>
<p>如果当前看到的中间元素的数值恰好等于目标元素，应该<strong>继续使用二分查找</strong>，查找它第一次出现和最后一次出现的位置，不可以线性地去扫描。</p>
</blockquote>
<h3 id="方法一：暴力求解（Brute-Force）"><a href="#方法一：暴力求解（Brute-Force）" class="headerlink" title="方法一：暴力求解（Brute Force）"></a>方法一：暴力求解（Brute Force）</h3><blockquote>
<p>该方法大家都会，可以跳过。</p>
</blockquote>
<p>很容易想到的一个方法是暴力解法，我们只需要从头到尾遍历一次数组，就能够找到目标元素出现的第一个位置和最后一个位置。根据题意，我们想象最一般的那种情况，先遇到的数都是严格小于 <code>target</code>，然后遇到的数都等于 <code>target</code>，最后遇到的数都严格大于  <code>target</code>。</p>
<ul>
<li>因此，我们可以在遍历的开始，检查遍历到的元素是否等于  <code>target</code> ，遇到刚好等于 <code>target</code> 的时候，记录当前的位置；</li>
<li>然后接着遍历，检查遍历到的元素是否不等于  <code>target</code> ，遇到刚好不等于 <code>target</code> 的时候，记录当前位置的前一个位置即可。</li>
</ul>
<p>这个算法的时间复杂度是 $O(N)$，不符合题目的要求。</p>
<p>Java 代码：</p>
<pre class="line-numbers language-java"><code class="language-java"><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">Solution</span> <span class="token punctuation">{</span>

    <span class="token keyword">public</span> <span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> <span class="token function">searchRange</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> nums<span class="token punctuation">,</span> <span class="token keyword">int</span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> targetRange <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span><span class="token punctuation">{</span><span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">}</span><span class="token punctuation">;</span>

        <span class="token keyword">int</span> len <span class="token operator">=</span> nums<span class="token punctuation">.</span>length<span class="token punctuation">;</span>
        <span class="token keyword">if</span> <span class="token punctuation">(</span>len <span class="token operator">==</span> <span class="token number">0</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">return</span> targetRange<span class="token punctuation">;</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> len<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">if</span> <span class="token punctuation">(</span>nums<span class="token punctuation">[</span>i<span class="token punctuation">]</span> <span class="token operator">==</span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
                targetRange<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span> <span class="token operator">=</span> i<span class="token punctuation">;</span>
                <span class="token keyword">break</span><span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>

        <span class="token comment" spellcheck="true">// 连第 1 个位置都没有找到，说明已经遍历完整个数组了</span>
        <span class="token keyword">if</span> <span class="token punctuation">(</span>targetRange<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span> <span class="token operator">==</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">return</span> targetRange<span class="token punctuation">;</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> targetRange<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span> <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> len<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">if</span> <span class="token punctuation">(</span>nums<span class="token punctuation">[</span>i<span class="token punctuation">]</span> <span class="token operator">!=</span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
                targetRange<span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">=</span> i <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span>
                <span class="token keyword">break</span><span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">if</span> <span class="token punctuation">(</span>targetRange<span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">==</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            targetRange<span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">=</span> len <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">return</span> targetRange<span class="token punctuation">;</span>
    <span class="token punctuation">}</span>
<span class="token punctuation">}</span><span aria-hidden="true" class="line-numbers-rows"><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span></span></code></pre>
<p><strong>代码讲解</strong>：</p>
<ul>
<li>首先新建一个有两个元素的整型数组 <code>targetRange</code>，初始化的值为 <code>[-1, -1]</code>，然后把数组的长度赋值成一个变量 <code>len</code>，当数组的长度等于 <code>0</code> 的时候，直接返回 <code>targetRange</code>；</li>
<li>然后从下标为 <code>0</code> 的地方开始遍历，只要找到了等于 <code>target</code> 的元素，就将 <code>targetRange</code> 下标为 <code>0</code> 的那个元素赋值为 <code>i</code> ，然后退出循环；</li>
<li>这里要注意的一点是：如果在整个遍历的过程中， <code>targetRange[0]</code> 都没有被重新赋值，那就说明目标元素 <code>target</code> 在有序数组 <code>nums</code> 中并不存在，依然将   <code>targetRange</code> 返回即可；</li>
<li>接着我们从目标元素第一次出现的位置  <code>targetRange[0]</code>  的下一个位置开始遍历，只要检测到元素不等于  <code>target</code> ，记录它的前一个位置，即将  <code>targetRange[1]</code> 赋值为 <code>i - 1</code> ，然后就可以退出循环；</li>
<li>还要注意的一个细节是，如果目标元素刚好在有序数组的最后一个位置，其实这个循环体是根本没有办法执行的，因此，可以在最后做一个判断，如果  <code>targetRange[1]</code>  还没有被赋值，就把它赋值为数组中最后一个位置。</li>
</ul>
<h3 id="方法二：二分查找"><a href="#方法二：二分查找" class="headerlink" title="方法二：二分查找"></a>方法二：二分查找</h3><p>下面我们看一下如何使用二分查找，找到目标元素在有序数组中的开始位置和结束位置。</p>
<ul>
<li><p>二分查找法的基本思想是：在一个区间范围里看处在中间位置的元素的值 <code>nums[mid]</code> 与目标元素 <code>target</code> 的大小关系，进而决定目标值落在哪一个部分里；</p>
</li>
<li><p>对于这道题，与常见的二分查找问题最大的不同就在于，目标元素  <code>target</code> 在有序数组中很可能存在多个；</p>
</li>
<li><p>而当我们使用二分查找方法看到的处在中间位置的元素的值 <code>nums[mid]</code> 恰好等于目标元素  <code>target</code> 的时候，还需要继续查找下去，而此时比较容易陷入的误区是线性查找，正确的做法是继续二分查找；</p>
</li>
<li><p>很显然，目标元素第 1 次出现的位置一定不可能是严格小于 <code>target</code> 的元素的位置，根据之前暴力解法的分析，目标元素第 1 次出现的位置是严格小于 <code>target</code> 的元素的位置的边界，因此，我们可以通过这个思路使用二分法找到这个边界；</p>
</li>
<li><p>对称地，目标元素最后 1 次出现的位置一定不可能是严格大于 <code>target</code> 的元素的位置，目标元素最后 1 次出现的位置是严格大于 <code>target</code> 的元素的位置的边界，因此，我们可以通过这个思路使用二分法找到这个边界。</p>
</li>
</ul>
<p>Java 代码：</p>
<pre class="line-numbers language-java"><code class="language-java"><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">Solution</span> <span class="token punctuation">{</span>

    <span class="token keyword">public</span> <span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> <span class="token function">searchRange</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> nums<span class="token punctuation">,</span> <span class="token keyword">int</span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">int</span> len <span class="token operator">=</span> nums<span class="token punctuation">.</span>length<span class="token punctuation">;</span>
        <span class="token keyword">if</span> <span class="token punctuation">(</span>len <span class="token operator">==</span> <span class="token number">0</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">return</span> <span class="token keyword">new</span> <span class="token class-name">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span><span class="token punctuation">{</span><span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">}</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">int</span> firstPosition <span class="token operator">=</span> <span class="token function">findFirstPosition</span><span class="token punctuation">(</span>nums<span class="token punctuation">,</span> target<span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token keyword">if</span> <span class="token punctuation">(</span>firstPosition <span class="token operator">==</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">return</span> <span class="token keyword">new</span> <span class="token class-name">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span><span class="token punctuation">{</span><span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">}</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">int</span> lastPosition <span class="token operator">=</span> <span class="token function">findLastPosition</span><span class="token punctuation">(</span>nums<span class="token punctuation">,</span> target<span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token keyword">return</span> <span class="token keyword">new</span> <span class="token class-name">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span><span class="token punctuation">{</span>firstPosition<span class="token punctuation">,</span> lastPosition<span class="token punctuation">}</span><span class="token punctuation">;</span>
    <span class="token punctuation">}</span>

    <span class="token keyword">private</span> <span class="token keyword">int</span> <span class="token function">findFirstPosition</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> nums<span class="token punctuation">,</span> <span class="token keyword">int</span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">int</span> left <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
        <span class="token keyword">int</span> right <span class="token operator">=</span> nums<span class="token punctuation">.</span>length <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span>
        <span class="token keyword">while</span> <span class="token punctuation">(</span>left <span class="token operator">&lt;</span> right<span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">int</span> mid <span class="token operator">=</span> <span class="token punctuation">(</span>left <span class="token operator">+</span> right<span class="token punctuation">)</span> <span class="token operator">>>></span> <span class="token number">1</span><span class="token punctuation">;</span>
            <span class="token comment" spellcheck="true">// 小于一定不是解</span>
            <span class="token keyword">if</span> <span class="token punctuation">(</span>nums<span class="token punctuation">[</span>mid<span class="token punctuation">]</span> <span class="token operator">&lt;</span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
                <span class="token comment" spellcheck="true">// 下一轮搜索区间是 [mid + 1, right]</span>
                left <span class="token operator">=</span> mid <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">;</span>
            <span class="token punctuation">}</span> <span class="token keyword">else</span> <span class="token punctuation">{</span>
                right <span class="token operator">=</span> mid<span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">if</span> <span class="token punctuation">(</span>nums<span class="token punctuation">[</span>left<span class="token punctuation">]</span> <span class="token operator">==</span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">return</span> left<span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">return</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">;</span>
    <span class="token punctuation">}</span>

    <span class="token keyword">private</span> <span class="token keyword">int</span> <span class="token function">findLastPosition</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> nums<span class="token punctuation">,</span> <span class="token keyword">int</span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">int</span> left <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
        <span class="token keyword">int</span> right <span class="token operator">=</span> nums<span class="token punctuation">.</span>length <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span>
        <span class="token keyword">while</span> <span class="token punctuation">(</span>left <span class="token operator">&lt;</span> right<span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">int</span> mid <span class="token operator">=</span> <span class="token punctuation">(</span>left <span class="token operator">+</span> right <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">)</span> <span class="token operator">>>></span> <span class="token number">1</span><span class="token punctuation">;</span>
            <span class="token comment" spellcheck="true">// 大于一定不是解</span>
            <span class="token keyword">if</span> <span class="token punctuation">(</span>nums<span class="token punctuation">[</span>mid<span class="token punctuation">]</span> <span class="token operator">></span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
                <span class="token comment" spellcheck="true">// 下一轮搜索区间是 [left, mid - 1]</span>
                right <span class="token operator">=</span> mid <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span>
            <span class="token punctuation">}</span> <span class="token keyword">else</span> <span class="token punctuation">{</span>
                left <span class="token operator">=</span> mid<span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">return</span> left<span class="token punctuation">;</span>
    <span class="token punctuation">}</span>
<span class="token punctuation">}</span><span aria-hidden="true" class="line-numbers-rows"><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span></span></code></pre>
<p><strong>代码讲解</strong>：</p>
<ul>
<li><p>首先依然是将数组的长度赋值给一个变量 <code>len</code>；</p>
</li>
<li><p>然后做一个特判：当数组的长度为 <code>0</code> 的时候，直接返回 <code>[-1 , -1]</code>；</p>
</li>
<li><p>接下来我们先二分查找 <code>target</code> 第 1 次出现的位置，我们把它封装成一个私有方法 <code>findFirstPosition</code>：</p>
<ul>
<li>分别初始化左边界 <code>left = 0</code>  和右边界 <code>right = len - 1</code> ，表示待搜索区间的左右边界；</li>
<li>接下来不断地二分查找目标元素，我们把循环可以继续的条件写成 <code>while (left &lt; right)</code> ，注意这里写成严格小于，这是一种比较常见的写法，它的思路是把符合条件的元素放在最后做判断，好处是在退出循环的时候一定有 <code>left == right</code> 成立，我们就不用考虑到底应该返回 <code>left</code> 还是返回 <code>right</code> 了。但是这种写法有几个要注意的地方，我们马上会提到。</li>
<li>然后是计算中间位置下标的代码 <code>int mid = left + (right - left) / 2;</code></li>
<li>根据刚才的分析，当 <code>nums[mid] &lt; target</code> 的时候，<code>mid</code> 以及 <code>mid</code> 左边的所有元素一定不是 <code>target</code> 出现的第 1 个位置，因此下一轮搜索的元素就一定在 <code>[mid + 1, right]</code> 中，因此，将左边界设置为 <code>mid + 1</code>；</li>
<li>接下来是 <code>nums[mid] &lt; target</code> 的反面， <code>nums[mid] &gt;= target</code> ，此时搜索的区间是 <code>if</code> 这个分支搜索区间的反面，即 <code>[left, mid]</code> ，我们来验证一下：如果我们看到一个数严格大于 <code>target</code> ， <code>target</code> 出现的第 1 个位置一定出现在这个数的左边；如果我们看到一个数恰好等于 <code>target</code> ，它有可能就是出现的第 1 个位置，也有可能 <code>target</code> 出现的第 1 个位置在它左边，但是一定不会在这个位置右边，因此，下一轮搜索的区间是  <code>[left, mid]</code> 是没有问题的，此时需要设置右边界 <code>right = mid</code>；</li>
<li>在退出循环的时候，还有一个下标的元素 <code>left</code>（或者说是 <code>right</code> 此时它们的值相等）没有看到，因为题目中说，目标元素有可能在数组中并不存在，因此，需要单独再做一次判断，这一步叫做后处理。</li>
<li>如果 <code>nums[left] == target</code> ，下标位置 <code>left</code> 就是 <code>target</code> 第 1 次出现的位置，否则返回 <code>-1</code>。</li>
</ul>
</li>
<li><p>对称地，我们来写查找 <code>target</code> 最后 1 次出现的位置的代码。</p>
<ul>
<li><p>首先可以确定的是，如果在查找 <code>target</code> 第 1 次出现的位置的时候，我们都没有找到  <code>target</code> ，在查找 <code>target</code>  最后一次出现的位置的时候，我们肯定也不会找到这个数，因此可以先做一个特殊的判断；</p>
</li>
<li><p><code>findLastPosition</code> 的结构和 <code>findFirstPosition</code> 很像，我们直接复制下来，需要更改的是 <code>if</code> 和 <code>else</code> 的逻辑，根据刚才的分析，当 <code>nums[mid] &gt; target</code> 的时候，<code>mid</code> 以及 <code>mid</code> 右边的所有元素一定不是 <code>target</code> 出现的最后 1 个位置，因此下一轮搜索的元素就一定在 <code>[left, mid - 1]</code> 中，因此，将右边界设置为 <code>mid - 1</code>；</p>
</li>
<li><p>接下来是 <code>nums[mid] &gt; target</code> 的反面， <code>nums[mid] &lt;= target</code> ，此时搜索的区间是 <code>if</code> 这个分支搜索区间的反面，即 <code>[mid, right]</code> ，我们来验证一下：如果我们看到一个数严格小于 <code>target</code> ， <code>target</code> 出现的最后 1 个位置一定出现在这个数的右边；如果我们看到一个数恰好等于 <code>target</code> ，它有可能就是出现的最后 1 个位置，也有可能 <code>target</code> 出现的最后 1 个位置在它右边，但是一定不会在这个位置左边，因此，下一轮搜索的区间是 <code>[mid, right]</code>  是没有问题的，此时需要设置左边界 <code>left = mid</code>；</p>
</li>
<li><p>此时要特别注意的一点是：一旦看到  <code>left = mid</code> 与 <code>right = mid - 1</code> 这种在二分搜索中边界收缩的行为，我们需要在取中间数的时候，做一些小的调整，那就是在这个括号里加 1：<code>int mid = left + (right - left + 1) / 2;</code></p>
</li>
<li><p>原因是这样的：<code>/</code> 是整数除法，它默认的取整行为是向下取整，即 <code>(3 + 4) / 2 = 3</code>。当我们使用原来写法的时候，<code>mid</code> 永远取不到 <code>right</code>，而边界收缩是  <code>left = mid</code> 与 <code>right = mid - 1</code> 的时候，我们画一个示意图，它是这样的，当待搜索区间只有 2 个元素的时候，由于中间数永远取不到右边，我们发现区间分不开，此时，一旦代码执行到这一个分支，无论是左边界还是右边界都不会向中间考虑，此时代码进入死循环；</p>
</li>
<li><p>要想解决这个问题，其实刚刚我已经说了，需要把取中间数时候整数除法的下取整行为改成上取整，即在括号里加 <code>1</code>，虽然这种取整行为的改变只需要在最后只剩 2 个元素的时候做出，但是我们全程就让它上取整也是没有问题的；</p>
</li>
<li><p>这一条是一个经验总结，也是人们在使用的过程中逐渐总结出来的，其实这个处理也不难发现，我们只需要在出现死循环的时候，在程序中打印出 <code>left</code> 、<code>right</code> 和 <code>mid</code> 的取值，就很容易发现问题并想到解决的办法；</p>
</li>
<li><p>在退出循环的时候，这里依然是还有一个下标的元素 <code>left</code>（或者说是 <code>right</code> 此时它们的值相等）没有看到。但是我们注意到，其实代码能够执行到这里，是不是说明，<code>target</code> 在 <code>nums</code> 中一定存在，看一看我们之前的判断。因此，我们就没有必要在判断  <code>nums[left]</code> 是否等于 <code>target</code>，此时 <code>left</code>的值就一定是 <code>target</code> 在有序数组 <code>nums</code> 中最后 1 次出现的下标 。</p>
</li>
</ul>
</li>
</ul>
<p>调试代码：</p>
<p>Java 代码：</p>
<pre class="line-numbers language-java"><code class="language-java"><span class="token keyword">class</span> <span class="token class-name">Solution</span> <span class="token punctuation">{</span>
    <span class="token keyword">public</span> <span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> <span class="token function">searchRange</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> nums<span class="token punctuation">,</span> <span class="token keyword">int</span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">int</span> len <span class="token operator">=</span> nums<span class="token punctuation">.</span>length<span class="token punctuation">;</span>
        <span class="token keyword">if</span> <span class="token punctuation">(</span>len <span class="token operator">==</span> <span class="token number">0</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">return</span> <span class="token keyword">new</span> <span class="token class-name">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span><span class="token punctuation">{</span><span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">}</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">int</span> firstPosition <span class="token operator">=</span> <span class="token function">findFirstPosition</span><span class="token punctuation">(</span>nums<span class="token punctuation">,</span> target<span class="token punctuation">,</span> len<span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token keyword">if</span> <span class="token punctuation">(</span>firstPosition <span class="token operator">==</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">return</span> <span class="token keyword">new</span> <span class="token class-name">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span><span class="token punctuation">{</span><span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">}</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">int</span> lastPosition <span class="token operator">=</span> <span class="token function">findLastPosition</span><span class="token punctuation">(</span>nums<span class="token punctuation">,</span> target<span class="token punctuation">,</span> len<span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token keyword">return</span> <span class="token keyword">new</span> <span class="token class-name">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span><span class="token punctuation">{</span>firstPosition<span class="token punctuation">,</span> lastPosition<span class="token punctuation">}</span><span class="token punctuation">;</span>
    <span class="token punctuation">}</span>

    <span class="token keyword">private</span> <span class="token keyword">int</span> <span class="token function">findLastPosition</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> nums<span class="token punctuation">,</span> <span class="token keyword">int</span> target<span class="token punctuation">,</span> <span class="token keyword">int</span> len<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">int</span> left <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
        <span class="token keyword">int</span> right <span class="token operator">=</span> len <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span>
        <span class="token keyword">while</span> <span class="token punctuation">(</span>left <span class="token operator">&lt;</span> right<span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">int</span> mid <span class="token operator">=</span> left <span class="token operator">+</span> <span class="token punctuation">(</span>right <span class="token operator">-</span> left<span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token number">2</span><span class="token punctuation">;</span>
            System<span class="token punctuation">.</span>out<span class="token punctuation">.</span><span class="token function">println</span><span class="token punctuation">(</span><span class="token string">"left = "</span> <span class="token operator">+</span> left <span class="token operator">+</span> <span class="token string">", mid = "</span> <span class="token operator">+</span> mid <span class="token operator">+</span> <span class="token string">", right = "</span> <span class="token operator">+</span> right<span class="token punctuation">)</span><span class="token punctuation">;</span>

            <span class="token comment" spellcheck="true">// 严格大于 target 的时候不是解</span>
            <span class="token keyword">if</span> <span class="token punctuation">(</span>nums<span class="token punctuation">[</span>mid<span class="token punctuation">]</span> <span class="token operator">></span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
                <span class="token comment" spellcheck="true">// 下一轮搜索的区间是 [left, mid - 1]</span>
                System<span class="token punctuation">.</span>out<span class="token punctuation">.</span><span class="token function">println</span><span class="token punctuation">(</span><span class="token string">"下一轮搜索的区间是 [left, mid - 1]"</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
                right <span class="token operator">=</span> mid <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span>
            <span class="token punctuation">}</span> <span class="token keyword">else</span> <span class="token punctuation">{</span>
                System<span class="token punctuation">.</span>out<span class="token punctuation">.</span><span class="token function">println</span><span class="token punctuation">(</span><span class="token string">"下一轮搜索的区间是 [mid, right]"</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
                <span class="token comment" spellcheck="true">// [mid, right]</span>
                left <span class="token operator">=</span> mid<span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">return</span> left<span class="token punctuation">;</span>
    <span class="token punctuation">}</span>

    <span class="token keyword">private</span> <span class="token keyword">int</span> <span class="token function">findFirstPosition</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> nums<span class="token punctuation">,</span> <span class="token keyword">int</span> target<span class="token punctuation">,</span> <span class="token keyword">int</span> len<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">int</span> left <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
        <span class="token keyword">int</span> right <span class="token operator">=</span> len <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span>
        <span class="token keyword">while</span> <span class="token punctuation">(</span>left <span class="token operator">&lt;</span> right<span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">int</span> mid <span class="token operator">=</span> left <span class="token operator">+</span> <span class="token punctuation">(</span>right <span class="token operator">-</span> left<span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token number">2</span><span class="token punctuation">;</span>
            <span class="token comment" spellcheck="true">// 严格小于 target 的时候不是解</span>
            <span class="token keyword">if</span> <span class="token punctuation">(</span>nums<span class="token punctuation">[</span>mid<span class="token punctuation">]</span> <span class="token operator">&lt;</span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
                <span class="token comment" spellcheck="true">// 下一轮搜索的区间是 [mid + 1, right]</span>
                left <span class="token operator">=</span> mid <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">;</span>
            <span class="token punctuation">}</span> <span class="token keyword">else</span> <span class="token punctuation">{</span>
                right <span class="token operator">=</span> mid<span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">if</span> <span class="token punctuation">(</span>nums<span class="token punctuation">[</span>left<span class="token punctuation">]</span> <span class="token operator">==</span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">return</span> left<span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">return</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">;</span>
    <span class="token punctuation">}</span>
<span class="token punctuation">}</span><span aria-hidden="true" class="line-numbers-rows"><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span></span></code></pre>
<p><strong>复杂度分析</strong>：</p>
<ul>
<li>时间复杂度：$O(\log N)$，这里 $N$ 是数组的长度，两个子问题都是二分查找，因此时间复杂度为对数级别。</li>
<li>空间复杂度：$O(1)$，只使用了常数个数的辅助变量、指针。</li>
</ul>
<p>（本节完）</p>

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                            「力扣」第 35 题：搜索插入位置（简单）
链接

题解链接（含视频讲解）



给定一个排序数组和一个目标值，在数组中找到目标值，并返回其索引。如果目标值不存在于数组中，返回它将会被按顺序插入的位置。
你可以假设数组中无重复元素。
示例 
                        
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                            「力扣」第 33 题：搜索旋转排序数组（中等）
链接

题解链接



假设按照升序排序的数组在预先未知的某个点上进行了旋转。
( 例如，数组 [0, 1, 2, 4, 5, 6, 7] 可能变为 [4, 5, 6, 7, 0, 1, 2]
                        
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